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x_1^k+x_2^k+...+x_n^k=1, (1+x_1)(1+x_2)...(1+x_n)=2

Source: Czech And Slovak Mathematical Olympiad, Round III, Category A 1982 p6

September 10, 2024
algebrasystem of equations

Problem Statement

Let n,kn,k be given natural numbers. Determine all ordered n-tuples of non-negative real numbers (x1,x2,...,xn)(x_1,x_2,...,x_n) that satisfy the system of equations x1k+x2k+...+xnk=1x_1^k+x_2^k+...+x_n^k=1 (1+x1)(1+x2)...(1+xn)=2(1+x_1)(1+x_2)...(1+x_n)=2
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